Aggregation Functions in Theory and in Practice

Frontmatter

Capacities, Survival Functions and Universal Integrals

Abstract

Based on the equality of survival functions related to n-ary vectors and capacities on \(X=\{1, \dots ,n \}\), the equality of universal integrals \({{\mathbf {I}}} (\mu ,{{\mathbf {x}}} )={{\mathbf {I}}} (\mu ,{{\mathbf {y}}} )\) is discussed and studied. Some particular cases are highlighted, and a special stress is put on possibility and necessity measures. As a by-product, a new characterization of possibility (necessity) measures is introduced.

Radko Mesiar, Andrea Stupňanová

Point-Interval-Valued Sets: Aggregation and Construction

Abstract

The concept of a point-interval-valued set (PIV set) is proposed as a tool for summary characterization of data from a two-way table. A PIV set is an L-fuzzy set whose membership labels can be numbers as well as special subintervals from the unit interval. Two relations of partial order of PIV sets are introduced and corresponding operations of union and intersection are studied. Aggregation of PIV sets by bounded t-norms is suggested.

Slavka Bodjanova, Martin Kalina

On Some Applications of Williamson’s Transform in Copula Theory

Abstract

We show several interesting examples of connection between distribution of a positively valued random variable and an Archimedean copula through Williamson’s transformation (and Laplace transform), especially when arranged in a sequence. Naturally, there appears a question: how can we use statistical properties of distance functions to draw statistical properties of copulas, and vice versa? This question is formulated in two open problems.

Tomáš Bacigál

Some Remarks on Idempotent Nullnorms on Bounded Lattices

Abstract

Nullnorms are generalizations of triangular norms (t-norms) and triangular conorms (t-conorms) with a zero element to be an arbitrary point from an arbitrary bounded lattice. In this paper, we study on the existence of idempotent nullnorms on bounded lattices. We show that there exists unique idempotent nullnorm on an arbitrary distributive bounded lattice. We prove that an idempotent nullnorm may not always exist on every bounded lattice. Furthermore, we propose the construction method to obtain idempotent nullnorms on a bounded lattice under additional assumptions on given zero element. As by-product of this method, we see that it is in existence an idempotent nullnorm on non-distributive bounded lattices.

Gül Deniz Çaylı, Funda Karaçal

Aggregating Fuzzy Subgroups and T-vague Groups

Abstract

Fuzzy subgroups and T-vague groups are interesting fuzzy algebraic structures that have been widely studied. While fuzzy subgroups fuzzify the concept of crisp subgroup, T-vague groups can be identified with quotient groups of a group by a normal fuzzy subgroup and there is a close relation between both structures and T-indistinguishability operators (fuzzy equivalence relations).

In this paper the functions that aggregate fuzzy subgroups and T-vague groups will be studied. The functions aggregating T-indistinguishability operators have been characterized [9] and the main result of this paper is that the functions aggregating T-indistinguishability operators coincide with the ones that aggregate fuzzy subgroups and T-vague groups. In particular, quasi-arithmetic means and some OWA operators aggregate them if the t-norm is continuous Archimedean.

D. Boixader, G. Mayor, J. Recasens

Families of Perturbation Copulas Generalizing the FGM Family and Their Relations to Dependence Measures

Abstract

In this paper we provide an extension of special parametric class of perturbations of an arbitrary copula (given in [3]) that represent a partial generalization of the FGM family of copulas for parameters from the unit interval. However the FGM family is defined for parameters from the interval \([-1,1]\). We present a construction of perturbations of an arbitrary copula also for parameters from the interval \([-1,0]\) so that together with the former family of perturbations of copulas we get a generalization of the FGM family for the whole interval \([-1,1]\). We also investigated the influence of the parameters of the introduced class of perturbations of copulas on several measures of dependence (Spearman’s rho, Blomqvist’s beta, Gini’s gamma, Kendall’s tau).

Jozef Komorník, Magdaléna Komorníková, Jana Kalická

k-maxitivity of Order-Preserving Homomorphisms of Lattices

Abstract

The concept of k-maxitivity for order-preserving homomorphisms between bounded lattices is introduced and discussed. As particular cases, k-maxitive capacities and aggregation functions are studied and exemplified.

Radko Mesiar, Anna Kolesárová

On Some Classes of RU-Implications Satisfying U-Modus Ponens

Abstract

The Modus Ponens property for fuzzy implication functions is essential in the inference process in approximate reasoning. It is usually considered with respect to a continuous t-norm T but it can be generalized to any conjunctor and, in particular, to a conjunctive uninorm U. In this paper, it is investigated when RU-implications derived from uninorms satisfy the Modus Ponens with respect to a conjunctive uninorm U. The new property, called here U-Modus Ponens, is studied in detail for RU-implications derived from uninorms lying in the classes of representable uninorms and uninorms continuous in the open unit square.

Margarita Mas, Daniel Ruiz-Aguilera, Joan Torrens

CMin-Integral: A Choquet-Like Aggregation Function Based on the Minimum t-Norm for Applications to Fuzzy Rule-Based Classification Systems

Abstract

This paper studies the concept of Choquet-like copula-based aggregation function (CC-integral), introduced by Lucca et al. [1], when one considers the Minimum t-norm, showing an application in fuzzy rule-based classification systems. The CC-integral is built from the standard Choquet integral, which is expanded by distributing the product operation, and, then, the product operation is generalized by a copula. In this paper, we study the behavior of this aggregation function in fuzzy rule-based classification systems, when one considers the Minimum t-norm as de copula of the CC-integral, which we call the CMin-integral. We show that the CMin-integral obtains a performance that is, with a high level of confidence, better than the approach that adopts the winning rule (maximum). Moreover, its behaviour is similar to the best Choquet-like pre-aggregation functions, introduced by Lucca et al. [10], with excellent performance. Consequently, the CMin-integral enlarge the scope of the applications by offering new possibilities for defining fuzzy reasoning methods with a similar gain in performance.

Graçaliz Pereira Dimuro, Giancarlo Lucca, José António Sanz, Humberto Bustince, Benjamín Bedregal

Directional and Ordered Directional Monotonicity of Mixture Functions

Abstract

In this contribution, we discuss the concepts of so-called fusion functions, pre-aggregation functions and their directional and ordered directional monotonicity in the context of mixture functions. Mixture functions represent a special class of weighted averaging functions whose weights are determined by continuous weighting functions which depend on the input values. They need not be monotone, in general. If they are monotone increasing, they also belong to the important class of aggregation functions. If the are directionally monotone, they belong to the class of pre-aggregation functions.

Currently there is increased interest in studying generalized forms of monotonicity such as weak, directional or ordered directional monotonicity due to their possible application in fields such classification or image processing.

This paper discusses properties of selected mixture functions with special emphasis on their directional and ordered directional monotonicity. The concept of directional and ordered directional monotonicity of mixture functions is investigated with respect to linear and quadratic weighting functions.

Jana Špirková, Gleb Beliakov, Humberto Bustince, Javier Fernández

Using Uninorms and Nullnorms to Modify Fuzzy Implication Functions

Abstract

In this comunication, some construction methods of fuzzy implication functions based on uninorms, nullnorms and fuzzy negations are presented. The main idea is to use these methods in order to obtain new implication functions from old ones in such a way that the obtained implication satisfies a desired property even if the old implication does not satisfy it. In this line, the paper focuses in the following three properties: the control of the decreasingness with respect to the first variable, the strong negation property and the property: \(I(x,N(x))=N(x)\). However, other properties could be also considered in the same way through the proposed methods.

Isabel Aguiló, Jaume Suñer, Joan Torrens

On the Aggregation of Zadeh’s Z-Numbers Based on Discrete Fuzzy Numbers

Abstract

The accurate modelling of natural language is one of the main goals in the theory of computing with words. Based on this idea, Zadeh in 2011, introduced the concept of Z-number which has a great potential not only from the theoretical point of view but also for many possible applications such as in economics, decision analysis, risk assessment, etc. Recently, the authors proposed a new vision of Zadeh’s Z-numbers based on discrete fuzzy numbers that simplifies the computations and maintains the flexibility of the original model from the linguistic point of view. Following with this novel interpretation, in this paper, algebraic structures in the set of Zadeh’s Z-numbers are studied. In this framework, we propose a method to construct aggregation functions from couples of discrete aggregation functions. In particular, t-norms and t-conorms are built. Finally, an application to reach a final decision on a decision making problem is given.

Sebastia Massanet, Juan Vicente Riera, Joan Torrens

Aggregation over Property-Based Preference Domains

Abstract

We provide an axiomatic characterization of preorders that are defined with respect to a set of properties. Moreover, it is proven that property-based posets are in natural correspondence with topological spaces. This paper propose also a characterization and a generalization of a Sugeno-type integral in our framework.

Marta Cardin

Generalization of Czogała-Drewniak Theorem for n-ary Semigroups

Abstract

We investigate n-ary semigroups as a natural generalization of binary semigroups. We refer it as a pair \((X,F_n)\), where X is a set and an n-associative function \(F_n:X^n\rightarrow X\) is defined on X. We show that if \(F_n\) is idempotent, n-associative function which is monotone in each of its variables, defined on an interval \(I\subset \mathbb {R}\) and has a neutral element, then \(F_n\) is combination of the minimum and maximum operation. Moreover we can characterize the n-ary semigroups \((I, F_n)\) where \(F_n\) has the previous properties.

Gergely Kiss, Gabor Somlai

On Idempotent Discrete Uninorms

Abstract

In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values. More precisely we first show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing in each variable. Then we show that, in this characterization, symmetry can be replaced with both bisymmetry and existence of a neutral element.

Miguel Couceiro, Jimmy Devillet, Jean-Luc Marichal

On the F-partial Order and Equivalence Classes of Nullnorms

Abstract

In this paper, we define the set \(I_{F}^{(x)}\), denoting the set of all incomparable elements with arbitrary but fixed \(x\in (0,1)\) according to F-partial order and this set is deeply investigated. Then, an equivalence relation on the class of nullnorms induced by a F-partial order is defined and discussed. Finally, we give an answer to a recently posed open problem.

Emel Aşıcı

A Generalization of the Gravitational Search Algorithm

Abstract

In this work we propose a generalization of the gravitational search algorithm where the product in the expression of the gravitational attraction force is replaced by more general functions. We study some conditions which ensure convergence of our proposal and we show that we recover a wide class of aggregation functions to replace the product.

Humberto Bustince, Maria Minárová, Javier Fernandez, Mikel Sesma-Sara, Cedric Marco-Detchart, Javier Ruiz-Aranguren

On Stability of Families for Improper Aggregation Operators

Abstract

This work extends the notion of consistency in terms of stability for Families of Aggregation Operators (FAO), as defined in previous works. The notion of stability proposed in this work, not only extends the previous one, but it can be applied to a wider set of FAOs, particularly, to those that we name here as Family of Improper Aggregation Operators (FIAO), or improper FAOs. When the aggregated value cannot be considered as a new item from the input, the present definition of consistency cannot be applied. This is usual in several areas, namely in the development of social, economic and political indexes, as far as the aggregation process typically yield a new and different concept from the input elements.

Pablo Olaso, Karina Rojas, Daniel Gómez, Javier Montero

Sizes, Super Level Measures and Integrals

Abstract

The concept of super level measures as a generalization of classical level measures is discussed and studied in detail. Following the developing of the theory of \(L_p\)-spaces introduced by non-additive integrals based on super level measures we discuss the integration theory modified by super level measures and we compare it with the classical approach.

Lenka Halčinová

Monotonicity in the Construction of Ordinal Sums of Fuzzy Implications

Abstract

In this contribution we discus the problem of monotonicity of intervals in the ordinal sums of fuzzy implication constructions. As a result, new ways of constructing of ordinal sums of fuzzy implications are obtained. These methods allow to adapt the value of fuzzy implication to specific requirements. For our new methods of construction, several sufficient properties for obtaining a fuzzy implication as a result are presented. Moreover, preservation of some properties of the ordinal sums are examined. Among others neutrality property, identity property, and ordering property are considered.

Michał Baczyński, Paweł Drygaś, Radko Mesiar

On the Visualization of Discrete Non-additive Measures

Abstract

Non-additive measures generalize additive measures, and have been utilized in several applications. They are used to represent different types of uncertainty and also to represent importance in data aggregation. As non-additive measures are set functions, the number of values to be considered grows exponentially. This makes difficult their definition but also their interpretation and understanding. In order to support understability, this paper explores the topic of visualizing discrete non-additive measures using node-link diagram representations.

Juhee Bae, Elio Ventocilla, Maria Riveiro, Vicenç Torra

Generating Recommendations in GDM with an Allocation of Information Granularity

Abstract

A Group decision making process is carried out when human beings jointly make an election from a possible collection of alternatives. Here, a question of importance is to avoid winners and losers, in the sense that the choice is not any more attributable to any single individual, but all group members contribute to the decision. For this reason, the agreement or consensus achieved among all the individuals should be as high as possible. In this contribution, a feedback mechanism is presented in order to increase the consensus achieved among the decision makers involved in this kind of problems. It is based on granular computing, which is utilized here to provide the necessary flexibility to increase the consensus. The feedback mechanism is able to deal with heterogeneous contexts, that is, contexts in which the decision makers have importance degrees considering their capacity or talent to handle the problem.

Francisco Javier Cabrerizo, Juan Antonio Morente-Molinera, Sergio Alonso, Ignacio Javier Pérez, Raquel Ureña, Enrique Herrera-Viedma

Aggregation Functions, Similarity and Fuzzy Measures

Abstract

We propose a new method for constructing fuzzy measures. This method is based on a fixed aggregation function A, similarity measure S and a vector \(\mathbf {x} \in [0,1]^n\). Some illustrative examples yielding parametric families of fuzzy measures are given, and some properties of our method are studied.

Surajit Borkotokey, Magdaléna Komorníková, Jun Li, Radko Mesiar

On the Construction of Associative, Commutative and Increasing Operations by Paving

Abstract

Bodjanova, Kalina and Král’ recently introduced a construction method, called paving, which enables to define a new associative, commutative and increasing operation from a given one and a discrete representable partial operation. As a matter of fact, not every discrete t-norm is representable, i.e. it can not always be generated by some additive generator, and this also holds for t-conorms and uninorms. Inspired by this fact and the method of paving, we construct some new associative, commutative and increasing operations on the unit interval from a t-norm on the unit interval and a discrete t-norm, t-superconorm, t-conorm or uninorm. Because of the duality between t-norms and t-conorms, we also define some operations from a t-conorm and a discrete t-norm, t-subnorm, t-conorm or uninorm.

Wenwen Zong, Yong Su, Hua-Wen Liu, Bernard De Baets

On Implication Operators

Abstract

Distributivity properties play an important role in fuzzy research. Based on the solution of the autodistributivity functional equations, we give a characterisation of two types of distributivity of fuzzy implication. Based on the mean disjunctive operator, the mean implication operator is introduced. Using the Pliant operators -where all operators have a common generator function- we show that some weakened properties of the fuzzy mean implications are valid. In the propositions we use the fixed point of the negation as a threshold. Finally, the generalized modus ponens is examined in this framework.

József Dombi

Some Results About Fuzzy Consequence Operators and Fuzzy Preorders Using Conjunctors

Abstract

The purpose of this paper is to study fuzzy operators induced by fuzzy relations and fuzzy relations induced by fuzzy operators. Many results are obtained about the relationship between \(*\)-preorders and fuzzy consequences operators for a fixed t-norm \(*\). We analyse these properties by considering a semi-copula (generalization of t-norm concept) instead of a t-norm. Moreover, we show that the conditions imposed cannot be relaxed. We have been able to prove some important results about the relationships between fuzzy relations and fuzzy operators in this more general context.

Carlos Bejines, María Jesús Chasco, Jorge Elorza, Susana Montes

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